On off-diagonal hypergraph Ramsey numbers
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913295863119872 |
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| author | Conlon, David Fox, Jacob Gunby, Benjamin He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques |
| author_facet | Conlon, David Fox, Jacob Gunby, Benjamin He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques |
| contents | A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{Ω_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02021 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On off-diagonal hypergraph Ramsey numbers Conlon, David Fox, Jacob Gunby, Benjamin He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques Combinatorics A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{Ω_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear. |
| title | On off-diagonal hypergraph Ramsey numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.02021 |